Preheat oven to 400ºF.
Roll out the cans of croissant sheets and separate into diamond shaped sections. Place a small handful of rotisserie chicken on each diamond, fold dough over and press ends together to create dumplings. Cut up any excess dough into strips.
Line a 9x13" baking dish with one layer of dumplings. Pour the can of cream of chicken soup over the layer of dumplings, followed by the can of cream of mushroom soup, and chicken broth.
Place strips in a crisscross pattern on the top of the dumplings and soup mixture, season dish with salt and ground black pepper and bake until the dumplings are tender and cooked through and the top is bubbly and brown, about 30 minutes.